Left Termination proof of /tmp/tmpXM1Toe/NJ1.pl

Left Termination of the query pattern
rev(g,a)
w.r.t. the given Prolog program could successfully be proven:

0 Prolog

↳1 PrologToPiTRSViaGraphTransformerProof (⇒, 280 ms)

↳2 PiTRS

↳3 DependencyPairsProof (⇔, 5 ms)

↳4 PiDP

↳5 DependencyGraphProof (⇔, 0 ms)

↳6 PiDP

↳7 UsableRulesProof (⇔, 0 ms)

↳8 PiDP

↳9 PiDPToQDPProof (⇒, 16 ms)

↳10 QDP

↳11 QDPSizeChangeProof (⇔, 0 ms)

↳12 YES

(0) Obligation:

Clauses:

rev(LS, RES) :- r1(LS, [], RES).
r1([], RES, RES).
r1(.(X, Xs), Accm, RES) :- r1(Xs, .(X, Accm), RES).

Query: rev(g,a)

(1) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

Transformed Prolog program to (Pi-)TRS.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
3 [label="A: rev(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
5 [label="5: rev(T1, T2), SCOPE: 1, CLAUSE: 0\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: r1(T5, [], T7)\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: r1(T5, [], T7), SCOPE: 2, CLAUSE: 1 | r1(T5, [], T7), SCOPE: 2, CLAUSE: 2\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: r1(T5, [], T7), SCOPE: 2, CLAUSE: 1\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: r1(T5, [], T7), SCOPE: 2, CLAUSE: 2\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: r1(T15, .(T14, []), T17)\n\nT14 is ground\nT15 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: r1(T15, .(T14, []), T17), SCOPE: 3, CLAUSE: 1 | r1(T15, .(T14, []), T17), SCOPE: 3, CLAUSE: 2\n\nT14 is ground\nT15 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: r1(T15, .(T14, []), T17), SCOPE: 3, CLAUSE: 1\n\nT14 is ground\nT15 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: r1(T15, .(T14, []), T17), SCOPE: 3, CLAUSE: 2\n\nT14 is ground\nT15 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: r1(T34, .(T33, .(T35, [])), T37)\n\nT33 is ground\nT34 is ground\nT35 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: r1(T34, .(T33, .(T35, [])), T37), SCOPE: 4, CLAUSE: 1 | r1(T34, .(T33, .(T35, [])), T37), SCOPE: 4, CLAUSE: 2\n\nT33 is ground\nT34 is ground\nT35 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: r1(T34, .(T33, .(T35, [])), T37), SCOPE: 4, CLAUSE: 1\n\nT33 is ground\nT34 is ground\nT35 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: r1(T34, .(T33, .(T35, [])), T37), SCOPE: 4, CLAUSE: 2\n\nT33 is ground\nT34 is ground\nT35 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
94 [label="94: r1(T63, .(T62, .(T64, .(T65, []))), T67)\n\nT62 is ground\nT63 is ground\nT64 is ground\nT65 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
95 [label="95: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
97 [label="97: r1(T63, .(T62, .(T64, .(T65, []))), T67), SCOPE: 5, CLAUSE: 1 | r1(T63, .(T62, .(T64, .(T65, []))), T67), SCOPE: 5, CLAUSE: 2\n\nT62 is ground\nT63 is ground\nT64 is ground\nT65 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
108 [label="108: r1(T63, .(T62, .(T64, .(T65, []))), T67), SCOPE: 5, CLAUSE: 1\n\nT62 is ground\nT63 is ground\nT64 is ground\nT65 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
109 [label="109: r1(T63, .(T62, .(T64, .(T65, []))), T67), SCOPE: 5, CLAUSE: 2\n\nT62 is ground\nT63 is ground\nT64 is ground\nT65 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
115 [label="115: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
116 [label="116: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
117 [label="117: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
123 [label="123: r1(T102, .(T101, .(T103, .(T104, .(T105, [])))), T107)\n\nT101 is ground\nT102 is ground\nT103 is ground\nT104 is ground\nT105 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
124 [label="124: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
136 [label="136: r1(T102, .(T101, .(T103, .(T104, .(T105, [])))), T107), SCOPE: 6, CLAUSE: 1 | r1(T102, .(T101, .(T103, .(T104, .(T105, [])))), T107), SCOPE: 6, CLAUSE: 2\n\nT101 is ground\nT102 is ground\nT103 is ground\nT104 is ground\nT105 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
137 [label="137: r1(T102, .(T101, .(T103, .(T104, .(T105, [])))), T107), SCOPE: 6, CLAUSE: 1\n\nT101 is ground\nT102 is ground\nT103 is ground\nT104 is ground\nT105 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
138 [label="138: r1(T102, .(T101, .(T103, .(T104, .(T105, [])))), T107), SCOPE: 6, CLAUSE: 2\n\nT101 is ground\nT102 is ground\nT103 is ground\nT104 is ground\nT105 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
142 [label="142: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
145 [label="145: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
146 [label="146: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
153 [label="153: r1(T151, .(T150, .(T152, .(T153, .(T154, .(T155, []))))), T157)\n\nT150 is ground\nT151 is ground\nT152 is ground\nT153 is ground\nT154 is ground\nT155 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
154 [label="154: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
160 [label="160: r1(T151, .(T150, .(T152, .(T153, .(T154, .(T155, []))))), T157), SCOPE: 7, CLAUSE: 1 | r1(T151, .(T150, .(T152, .(T153, .(T154, .(T155, []))))), T157), SCOPE: 7, CLAUSE: 2\n\nT150 is ground\nT151 is ground\nT152 is ground\nT153 is ground\nT154 is ground\nT155 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
162 [label="162: r1(T151, .(T150, .(T152, .(T153, .(T154, .(T155, []))))), T157), SCOPE: 7, CLAUSE: 1\n\nT150 is ground\nT151 is ground\nT152 is ground\nT153 is ground\nT154 is ground\nT155 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
163 [label="163: r1(T151, .(T150, .(T152, .(T153, .(T154, .(T155, []))))), T157), SCOPE: 7, CLAUSE: 2\n\nT150 is ground\nT151 is ground\nT152 is ground\nT153 is ground\nT154 is ground\nT155 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
169 [label="169: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
170 [label="170: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
171 [label="171: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
176 [label="176: r1(T210, .(T209, .(T211, .(T212, .(T213, .(T214, .(T215, [])))))), T217)\n\nT209 is ground\nT210 is ground\nT211 is ground\nT212 is ground\nT213 is ground\nT214 is ground\nT215 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
177 [label="177: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
183 [label="183: r1(T210, .(T209, .(T211, .(T212, .(T213, .(T214, .(T215, [])))))), T217), SCOPE: 8, CLAUSE: 1 | r1(T210, .(T209, .(T211, .(T212, .(T213, .(T214, .(T215, [])))))), T217), SCOPE: 8, CLAUSE: 2\n\nT209 is ground\nT210 is ground\nT211 is ground\nT212 is ground\nT213 is ground\nT214 is ground\nT215 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
186 [label="186: r1(T210, .(T209, .(T211, .(T212, .(T213, .(T214, .(T215, [])))))), T217), SCOPE: 8, CLAUSE: 1\n\nT209 is ground\nT210 is ground\nT211 is ground\nT212 is ground\nT213 is ground\nT214 is ground\nT215 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
187 [label="187: r1(T210, .(T209, .(T211, .(T212, .(T213, .(T214, .(T215, [])))))), T217), SCOPE: 8, CLAUSE: 2\n\nT209 is ground\nT210 is ground\nT211 is ground\nT212 is ground\nT213 is ground\nT214 is ground\nT215 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
191 [label="191: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
192 [label="192: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
193 [label="193: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
199 [label="199: r1(T279, .(T278, .(T280, .(T281, .(T282, .(T283, .(T284, .(T285, []))))))), T287)\n\nT278 is ground\nT279 is ground\nT280 is ground\nT281 is ground\nT282 is ground\nT283 is ground\nT284 is ground\nT285 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
200 [label="200: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
201 [label="201: r1(T279, .(T278, .(T280, .(T281, .(T282, .(T283, .(T284, .(T285, []))))))), T287), SCOPE: 9, CLAUSE: 1 | r1(T279, .(T278, .(T280, .(T281, .(T282, .(T283, .(T284, .(T285, []))))))), T287), SCOPE: 9, CLAUSE: 2\n\nT278 is ground\nT279 is ground\nT280 is ground\nT281 is ground\nT282 is ground\nT283 is ground\nT284 is ground\nT285 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
202 [label="202: r1(T279, .(T278, .(T280, .(T281, .(T282, .(T283, .(T284, .(T285, []))))))), T287), SCOPE: 9, CLAUSE: 1\n\nT278 is ground\nT279 is ground\nT280 is ground\nT281 is ground\nT282 is ground\nT283 is ground\nT284 is ground\nT285 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
203 [label="203: r1(T279, .(T278, .(T280, .(T281, .(T282, .(T283, .(T284, .(T285, []))))))), T287), SCOPE: 9, CLAUSE: 2\n\nT278 is ground\nT279 is ground\nT280 is ground\nT281 is ground\nT282 is ground\nT283 is ground\nT284 is ground\nT285 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
213 [label="213: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
214 [label="214: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
215 [label="215: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
218 [label="218: r1(T358, .(T357, .(T359, .(T360, .(T361, .(T362, .(T363, .(T364, .(T365, [])))))))), T367)\n\nT357 is ground\nT358 is ground\nT359 is ground\nT360 is ground\nT361 is ground\nT362 is ground\nT363 is ground\nT364 is ground\nT365 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
219 [label="219: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
220 [label="B: r1(T358, .(T357, T384), T367)\n\nT357 is ground\nT358 is ground\nT384 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
221 [label="221: r1(T358, .(T357, T384), T367), SCOPE: 10, CLAUSE: 1 | r1(T358, .(T357, T384), T367), SCOPE: 10, CLAUSE: 2\n\nT357 is ground\nT358 is ground\nT384 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
222 [label="222: r1(T358, .(T357, T384), T367), SCOPE: 10, CLAUSE: 1\n\nT357 is ground\nT358 is ground\nT384 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
223 [label="223: r1(T358, .(T357, T384), T367), SCOPE: 10, CLAUSE: 2\n\nT357 is ground\nT358 is ground\nT384 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
224 [label="224: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
225 [label="225: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
226 [label="226: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
227 [label="227: r1(T410, .(T409, .(T411, T412)), T414)\n\nT409 is ground\nT410 is ground\nT411 is ground\nT412 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
228 [label="228: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
229 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 229 [arrowhead = none ];
229 -> 5;

230 [label="ONLY EVAL with clause\nrev(X3, X4) :- r1(X3, [], X4).\nand substitutionT1 -> T5,\nX3 -> T5,\nT2 -> T7,\nX4 -> T7,\nT6 -> T7", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 230 [arrowhead = none ];
230 -> 7;

231 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
7 -> 231 [arrowhead = none ];
231 -> 11;

232 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
11 -> 232 [arrowhead = none ];
232 -> 12;

233 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
11 -> 233 [arrowhead = none ];
233 -> 14;

234 [label="EVAL with clause\nr1([], X9, X9).\nand substitutionT5 -> [],\nX9 -> [],\nT7 -> []", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 234 [arrowhead = none ];
234 -> 17;

235 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
12 -> 235 [arrowhead = none ];
235 -> 19;

236 [label="EVAL with clause\nr1(.(X18, X19), X20, X21) :- r1(X19, .(X18, X20), X21).\nand substitutionX18 -> T14,\nX19 -> T15,\nT5 -> .(T14, T15),\nX20 -> [],\nT7 -> T17,\nX21 -> T17,\nT16 -> T17", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 236 [arrowhead = none ];
236 -> 29;

237 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
14 -> 237 [arrowhead = none ];
237 -> 31;

238 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 238 [arrowhead = none ];
238 -> 20;

239 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
29 -> 239 [arrowhead = none ];
239 -> 33;

240 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
33 -> 240 [arrowhead = none ];
240 -> 37;

241 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
33 -> 241 [arrowhead = none ];
241 -> 38;

242 [label="EVAL with clause\nr1([], X28, X28).\nand substitutionT15 -> [],\nT14 -> T24,\nX28 -> .(T24, []),\nT17 -> .(T24, [])", fontsize=14, style = filled, fillcolor = lightblue];
37 -> 242 [arrowhead = none ];
242 -> 42;

243 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
37 -> 243 [arrowhead = none ];
243 -> 45;

244 [label="EVAL with clause\nr1(.(X37, X38), X39, X40) :- r1(X38, .(X37, X39), X40).\nand substitutionX37 -> T33,\nX38 -> T34,\nT15 -> .(T33, T34),\nT14 -> T35,\nX39 -> .(T35, []),\nT17 -> T37,\nX40 -> T37,\nT36 -> T37", fontsize=14, style = filled, fillcolor = lightblue];
38 -> 244 [arrowhead = none ];
244 -> 57;

245 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
38 -> 245 [arrowhead = none ];
245 -> 58;

246 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
42 -> 246 [arrowhead = none ];
246 -> 46;

247 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
57 -> 247 [arrowhead = none ];
247 -> 69;

248 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
69 -> 248 [arrowhead = none ];
248 -> 73;

249 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
69 -> 249 [arrowhead = none ];
249 -> 74;

250 [label="EVAL with clause\nr1([], X47, X47).\nand substitutionT34 -> [],\nT33 -> T50,\nT35 -> T51,\nX47 -> .(T50, .(T51, [])),\nT37 -> .(T50, .(T51, []))", fontsize=14, style = filled, fillcolor = lightblue];
73 -> 250 [arrowhead = none ];
250 -> 76;

251 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
73 -> 251 [arrowhead = none ];
251 -> 77;

252 [label="EVAL with clause\nr1(.(X56, X57), X58, X59) :- r1(X57, .(X56, X58), X59).\nand substitutionX56 -> T62,\nX57 -> T63,\nT34 -> .(T62, T63),\nT33 -> T64,\nT35 -> T65,\nX58 -> .(T64, .(T65, [])),\nT37 -> T67,\nX59 -> T67,\nT66 -> T67", fontsize=14, style = filled, fillcolor = lightblue];
74 -> 252 [arrowhead = none ];
252 -> 94;

253 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
74 -> 253 [arrowhead = none ];
253 -> 95;

254 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
76 -> 254 [arrowhead = none ];
254 -> 78;

255 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
94 -> 255 [arrowhead = none ];
255 -> 97;

256 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
97 -> 256 [arrowhead = none ];
256 -> 108;

257 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
97 -> 257 [arrowhead = none ];
257 -> 109;

258 [label="EVAL with clause\nr1([], X66, X66).\nand substitutionT63 -> [],\nT62 -> T86,\nT64 -> T87,\nT65 -> T88,\nX66 -> .(T86, .(T87, .(T88, []))),\nT67 -> .(T86, .(T87, .(T88, [])))", fontsize=14, style = filled, fillcolor = lightblue];
108 -> 258 [arrowhead = none ];
258 -> 115;

259 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
108 -> 259 [arrowhead = none ];
259 -> 116;

260 [label="EVAL with clause\nr1(.(X75, X76), X77, X78) :- r1(X76, .(X75, X77), X78).\nand substitutionX75 -> T101,\nX76 -> T102,\nT63 -> .(T101, T102),\nT62 -> T103,\nT64 -> T104,\nT65 -> T105,\nX77 -> .(T103, .(T104, .(T105, []))),\nT67 -> T107,\nX78 -> T107,\nT106 -> T107", fontsize=14, style = filled, fillcolor = lightblue];
109 -> 260 [arrowhead = none ];
260 -> 123;

261 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
109 -> 261 [arrowhead = none ];
261 -> 124;

262 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
115 -> 262 [arrowhead = none ];
262 -> 117;

263 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
123 -> 263 [arrowhead = none ];
263 -> 136;

264 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
136 -> 264 [arrowhead = none ];
264 -> 137;

265 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
136 -> 265 [arrowhead = none ];
265 -> 138;

266 [label="EVAL with clause\nr1([], X85, X85).\nand substitutionT102 -> [],\nT101 -> T132,\nT103 -> T133,\nT104 -> T134,\nT105 -> T135,\nX85 -> .(T132, .(T133, .(T134, .(T135, [])))),\nT107 -> .(T132, .(T133, .(T134, .(T135, []))))", fontsize=14, style = filled, fillcolor = lightblue];
137 -> 266 [arrowhead = none ];
266 -> 142;

267 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
137 -> 267 [arrowhead = none ];
267 -> 145;

268 [label="EVAL with clause\nr1(.(X94, X95), X96, X97) :- r1(X95, .(X94, X96), X97).\nand substitutionX94 -> T150,\nX95 -> T151,\nT102 -> .(T150, T151),\nT101 -> T152,\nT103 -> T153,\nT104 -> T154,\nT105 -> T155,\nX96 -> .(T152, .(T153, .(T154, .(T155, [])))),\nT107 -> T157,\nX97 -> T157,\nT156 -> T157", fontsize=14, style = filled, fillcolor = lightblue];
138 -> 268 [arrowhead = none ];
268 -> 153;

269 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
138 -> 269 [arrowhead = none ];
269 -> 154;

270 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
142 -> 270 [arrowhead = none ];
270 -> 146;

271 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
153 -> 271 [arrowhead = none ];
271 -> 160;

272 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
160 -> 272 [arrowhead = none ];
272 -> 162;

273 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
160 -> 273 [arrowhead = none ];
273 -> 163;

274 [label="EVAL with clause\nr1([], X104, X104).\nand substitutionT151 -> [],\nT150 -> T188,\nT152 -> T189,\nT153 -> T190,\nT154 -> T191,\nT155 -> T192,\nX104 -> .(T188, .(T189, .(T190, .(T191, .(T192, []))))),\nT157 -> .(T188, .(T189, .(T190, .(T191, .(T192, [])))))", fontsize=14, style = filled, fillcolor = lightblue];
162 -> 274 [arrowhead = none ];
274 -> 169;

275 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
162 -> 275 [arrowhead = none ];
275 -> 170;

276 [label="EVAL with clause\nr1(.(X113, X114), X115, X116) :- r1(X114, .(X113, X115), X116).\nand substitutionX113 -> T209,\nX114 -> T210,\nT151 -> .(T209, T210),\nT150 -> T211,\nT152 -> T212,\nT153 -> T213,\nT154 -> T214,\nT155 -> T215,\nX115 -> .(T211, .(T212, .(T213, .(T214, .(T215, []))))),\nT157 -> T217,\nX116 -> T217,\nT216 -> T217", fontsize=14, style = filled, fillcolor = lightblue];
163 -> 276 [arrowhead = none ];
276 -> 176;

277 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
163 -> 277 [arrowhead = none ];
277 -> 177;

278 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
169 -> 278 [arrowhead = none ];
278 -> 171;

279 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
176 -> 279 [arrowhead = none ];
279 -> 183;

280 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
183 -> 280 [arrowhead = none ];
280 -> 186;

281 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
183 -> 281 [arrowhead = none ];
281 -> 187;

282 [label="EVAL with clause\nr1([], X123, X123).\nand substitutionT210 -> [],\nT209 -> T254,\nT211 -> T255,\nT212 -> T256,\nT213 -> T257,\nT214 -> T258,\nT215 -> T259,\nX123 -> .(T254, .(T255, .(T256, .(T257, .(T258, .(T259, [])))))),\nT217 -> .(T254, .(T255, .(T256, .(T257, .(T258, .(T259, []))))))", fontsize=14, style = filled, fillcolor = lightblue];
186 -> 282 [arrowhead = none ];
282 -> 191;

283 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
186 -> 283 [arrowhead = none ];
283 -> 192;

284 [label="EVAL with clause\nr1(.(X132, X133), X134, X135) :- r1(X133, .(X132, X134), X135).\nand substitutionX132 -> T278,\nX133 -> T279,\nT210 -> .(T278, T279),\nT209 -> T280,\nT211 -> T281,\nT212 -> T282,\nT213 -> T283,\nT214 -> T284,\nT215 -> T285,\nX134 -> .(T280, .(T281, .(T282, .(T283, .(T284, .(T285, [])))))),\nT217 -> T287,\nX135 -> T287,\nT286 -> T287", fontsize=14, style = filled, fillcolor = lightblue];
187 -> 284 [arrowhead = none ];
284 -> 199;

285 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
187 -> 285 [arrowhead = none ];
285 -> 200;

286 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
191 -> 286 [arrowhead = none ];
286 -> 193;

287 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
199 -> 287 [arrowhead = none ];
287 -> 201;

288 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
201 -> 288 [arrowhead = none ];
288 -> 202;

289 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
201 -> 289 [arrowhead = none ];
289 -> 203;

290 [label="EVAL with clause\nr1([], X142, X142).\nand substitutionT279 -> [],\nT278 -> T330,\nT280 -> T331,\nT281 -> T332,\nT282 -> T333,\nT283 -> T334,\nT284 -> T335,\nT285 -> T336,\nX142 -> .(T330, .(T331, .(T332, .(T333, .(T334, .(T335, .(T336, []))))))),\nT287 -> .(T330, .(T331, .(T332, .(T333, .(T334, .(T335, .(T336, [])))))))", fontsize=14, style = filled, fillcolor = lightblue];
202 -> 290 [arrowhead = none ];
290 -> 213;

291 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
202 -> 291 [arrowhead = none ];
291 -> 214;

292 [label="EVAL with clause\nr1(.(X151, X152), X153, X154) :- r1(X152, .(X151, X153), X154).\nand substitutionX151 -> T357,\nX152 -> T358,\nT279 -> .(T357, T358),\nT278 -> T359,\nT280 -> T360,\nT281 -> T361,\nT282 -> T362,\nT283 -> T363,\nT284 -> T364,\nT285 -> T365,\nX153 -> .(T359, .(T360, .(T361, .(T362, .(T363, .(T364, .(T365, []))))))),\nT287 -> T367,\nX154 -> T367,\nT366 -> T367", fontsize=14, style = filled, fillcolor = lightblue];
203 -> 292 [arrowhead = none ];
292 -> 218;

293 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
203 -> 293 [arrowhead = none ];
293 -> 219;

294 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
213 -> 294 [arrowhead = none ];
294 -> 215;

295 [label="GENERALIZATION\nT384 <-- .(T359, .(T360, .(T361, .(T362, .(T363, .(T364, .(T365, [])))))))\n\nNew Knowledge:\nT384 is ground", fontsize=14, style = filled, fillcolor = violet];
218 -> 295 [arrowhead = none ];
295 -> 220;

296 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
220 -> 296 [arrowhead = none ];
296 -> 221;

297 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
221 -> 297 [arrowhead = none ];
297 -> 222;

298 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
221 -> 298 [arrowhead = none ];
298 -> 223;

299 [label="EVAL with clause\nr1([], X163, X163).\nand substitutionT358 -> [],\nT357 -> T397,\nT384 -> T398,\nX163 -> .(T397, T398),\nT367 -> .(T397, T398)", fontsize=14, style = filled, fillcolor = lightblue];
222 -> 299 [arrowhead = none ];
299 -> 224;

300 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
222 -> 300 [arrowhead = none ];
300 -> 225;

301 [label="EVAL with clause\nr1(.(X172, X173), X174, X175) :- r1(X173, .(X172, X174), X175).\nand substitutionX172 -> T409,\nX173 -> T410,\nT358 -> .(T409, T410),\nT357 -> T411,\nT384 -> T412,\nX174 -> .(T411, T412),\nT367 -> T414,\nX175 -> T414,\nT413 -> T414", fontsize=14, style = filled, fillcolor = lightblue];
223 -> 301 [arrowhead = none ];
301 -> 227;

302 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
223 -> 302 [arrowhead = none ];
302 -> 228;

303 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
224 -> 303 [arrowhead = none ];
303 -> 226;

304 [label="INSTANCE with matching:\nT358 -> T410\nT357 -> T409\nT384 -> .(T411, T412)\nT367 -> T414", fontsize=14, style = filled, fillcolor = lightgrey];
227 -> 304 [arrowhead = none , style=dashed];
304 -> 220[style=dashed];

}

(2) Obligation:

Pi-finite rewrite system:
The TRS R consists of the following rules:

revA_in_ga([], []) → revA_out_ga([], [])
revA_in_ga(.(T24, []), .(T24, [])) → revA_out_ga(.(T24, []), .(T24, []))
revA_in_ga(.(T51, .(T50, [])), .(T50, .(T51, []))) → revA_out_ga(.(T51, .(T50, [])), .(T50, .(T51, [])))
revA_in_ga(.(T88, .(T87, .(T86, []))), .(T86, .(T87, .(T88, [])))) → revA_out_ga(.(T88, .(T87, .(T86, []))), .(T86, .(T87, .(T88, []))))
revA_in_ga(.(T135, .(T134, .(T133, .(T132, [])))), .(T132, .(T133, .(T134, .(T135, []))))) → revA_out_ga(.(T135, .(T134, .(T133, .(T132, [])))), .(T132, .(T133, .(T134, .(T135, [])))))
revA_in_ga(.(T192, .(T191, .(T190, .(T189, .(T188, []))))), .(T188, .(T189, .(T190, .(T191, .(T192, [])))))) → revA_out_ga(.(T192, .(T191, .(T190, .(T189, .(T188, []))))), .(T188, .(T189, .(T190, .(T191, .(T192, []))))))
revA_in_ga(.(T259, .(T258, .(T257, .(T256, .(T255, .(T254, [])))))), .(T254, .(T255, .(T256, .(T257, .(T258, .(T259, []))))))) → revA_out_ga(.(T259, .(T258, .(T257, .(T256, .(T255, .(T254, [])))))), .(T254, .(T255, .(T256, .(T257, .(T258, .(T259, [])))))))
revA_in_ga(.(T336, .(T335, .(T334, .(T333, .(T332, .(T331, .(T330, []))))))), .(T330, .(T331, .(T332, .(T333, .(T334, .(T335, .(T336, [])))))))) → revA_out_ga(.(T336, .(T335, .(T334, .(T333, .(T332, .(T331, .(T330, []))))))), .(T330, .(T331, .(T332, .(T333, .(T334, .(T335, .(T336, []))))))))
revA_in_ga(.(T365, .(T364, .(T363, .(T362, .(T361, .(T360, .(T359, .(T357, T358)))))))), T367) → U1_ga(T365, T364, T363, T362, T361, T360, T359, T357, T358, T367, r1B_in_ggga(T358, T357, .(T359, .(T360, .(T361, .(T362, .(T363, .(T364, .(T365, []))))))), T367))
r1B_in_ggga([], T397, T398, .(T397, T398)) → r1B_out_ggga([], T397, T398, .(T397, T398))
r1B_in_ggga(.(T409, T410), T411, T412, T414) → U2_ggga(T409, T410, T411, T412, T414, r1B_in_ggga(T410, T409, .(T411, T412), T414))
U2_ggga(T409, T410, T411, T412, T414, r1B_out_ggga(T410, T409, .(T411, T412), T414)) → r1B_out_ggga(.(T409, T410), T411, T412, T414)
U1_ga(T365, T364, T363, T362, T361, T360, T359, T357, T358, T367, r1B_out_ggga(T358, T357, .(T359, .(T360, .(T361, .(T362, .(T363, .(T364, .(T365, []))))))), T367)) → revA_out_ga(.(T365, .(T364, .(T363, .(T362, .(T361, .(T360, .(T359, .(T357, T358)))))))), T367)

(3) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

REVA_IN_GA(.(T365, .(T364, .(T363, .(T362, .(T361, .(T360, .(T359, .(T357, T358)))))))), T367) → U1_GA(T365, T364, T363, T362, T361, T360, T359, T357, T358, T367, r1B_in_ggga(T358, T357, .(T359, .(T360, .(T361, .(T362, .(T363, .(T364, .(T365, []))))))), T367))
REVA_IN_GA(.(T365, .(T364, .(T363, .(T362, .(T361, .(T360, .(T359, .(T357, T358)))))))), T367) → R1B_IN_GGGA(T358, T357, .(T359, .(T360, .(T361, .(T362, .(T363, .(T364, .(T365, []))))))), T367)
R1B_IN_GGGA(.(T409, T410), T411, T412, T414) → U2_GGGA(T409, T410, T411, T412, T414, r1B_in_ggga(T410, T409, .(T411, T412), T414))
R1B_IN_GGGA(.(T409, T410), T411, T412, T414) → R1B_IN_GGGA(T410, T409, .(T411, T412), T414)

The TRS R consists of the following rules:

revA_in_ga([], []) → revA_out_ga([], [])
revA_in_ga(.(T24, []), .(T24, [])) → revA_out_ga(.(T24, []), .(T24, []))
revA_in_ga(.(T51, .(T50, [])), .(T50, .(T51, []))) → revA_out_ga(.(T51, .(T50, [])), .(T50, .(T51, [])))
revA_in_ga(.(T88, .(T87, .(T86, []))), .(T86, .(T87, .(T88, [])))) → revA_out_ga(.(T88, .(T87, .(T86, []))), .(T86, .(T87, .(T88, []))))
revA_in_ga(.(T135, .(T134, .(T133, .(T132, [])))), .(T132, .(T133, .(T134, .(T135, []))))) → revA_out_ga(.(T135, .(T134, .(T133, .(T132, [])))), .(T132, .(T133, .(T134, .(T135, [])))))
revA_in_ga(.(T192, .(T191, .(T190, .(T189, .(T188, []))))), .(T188, .(T189, .(T190, .(T191, .(T192, [])))))) → revA_out_ga(.(T192, .(T191, .(T190, .(T189, .(T188, []))))), .(T188, .(T189, .(T190, .(T191, .(T192, []))))))
revA_in_ga(.(T259, .(T258, .(T257, .(T256, .(T255, .(T254, [])))))), .(T254, .(T255, .(T256, .(T257, .(T258, .(T259, []))))))) → revA_out_ga(.(T259, .(T258, .(T257, .(T256, .(T255, .(T254, [])))))), .(T254, .(T255, .(T256, .(T257, .(T258, .(T259, [])))))))
revA_in_ga(.(T336, .(T335, .(T334, .(T333, .(T332, .(T331, .(T330, []))))))), .(T330, .(T331, .(T332, .(T333, .(T334, .(T335, .(T336, [])))))))) → revA_out_ga(.(T336, .(T335, .(T334, .(T333, .(T332, .(T331, .(T330, []))))))), .(T330, .(T331, .(T332, .(T333, .(T334, .(T335, .(T336, []))))))))
revA_in_ga(.(T365, .(T364, .(T363, .(T362, .(T361, .(T360, .(T359, .(T357, T358)))))))), T367) → U1_ga(T365, T364, T363, T362, T361, T360, T359, T357, T358, T367, r1B_in_ggga(T358, T357, .(T359, .(T360, .(T361, .(T362, .(T363, .(T364, .(T365, []))))))), T367))
r1B_in_ggga([], T397, T398, .(T397, T398)) → r1B_out_ggga([], T397, T398, .(T397, T398))
r1B_in_ggga(.(T409, T410), T411, T412, T414) → U2_ggga(T409, T410, T411, T412, T414, r1B_in_ggga(T410, T409, .(T411, T412), T414))
U2_ggga(T409, T410, T411, T412, T414, r1B_out_ggga(T410, T409, .(T411, T412), T414)) → r1B_out_ggga(.(T409, T410), T411, T412, T414)
U1_ga(T365, T364, T363, T362, T361, T360, T359, T357, T358, T367, r1B_out_ggga(T358, T357, .(T359, .(T360, .(T361, .(T362, .(T363, .(T364, .(T365, []))))))), T367)) → revA_out_ga(.(T365, .(T364, .(T363, .(T362, .(T361, .(T360, .(T359, .(T357, T358)))))))), T367)

The argument filtering Pi contains the following mapping:
revA_in_ga(x1, x2) = revA_in_ga(x1)
[] = []
revA_out_ga(x1, x2) = revA_out_ga(x1, x2)
.(x1, x2) = .(x1, x2)
U1_ga(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U1_ga(x1, x2, x3, x4, x5, x6, x7, x8, x9, x11)
r1B_in_ggga(x1, x2, x3, x4) = r1B_in_ggga(x1, x2, x3)
r1B_out_ggga(x1, x2, x3, x4) = r1B_out_ggga(x1, x2, x3, x4)
U2_ggga(x1, x2, x3, x4, x5, x6) = U2_ggga(x1, x2, x3, x4, x6)
REVA_IN_GA(x1, x2) = REVA_IN_GA(x1)
U1_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U1_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x11)
R1B_IN_GGGA(x1, x2, x3, x4) = R1B_IN_GGGA(x1, x2, x3)
U2_GGGA(x1, x2, x3, x4, x5, x6) = U2_GGGA(x1, x2, x3, x4, x6)

We have to consider all (P,R,Pi)-chains

(4) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

REVA_IN_GA(.(T365, .(T364, .(T363, .(T362, .(T361, .(T360, .(T359, .(T357, T358)))))))), T367) → U1_GA(T365, T364, T363, T362, T361, T360, T359, T357, T358, T367, r1B_in_ggga(T358, T357, .(T359, .(T360, .(T361, .(T362, .(T363, .(T364, .(T365, []))))))), T367))
REVA_IN_GA(.(T365, .(T364, .(T363, .(T362, .(T361, .(T360, .(T359, .(T357, T358)))))))), T367) → R1B_IN_GGGA(T358, T357, .(T359, .(T360, .(T361, .(T362, .(T363, .(T364, .(T365, []))))))), T367)
R1B_IN_GGGA(.(T409, T410), T411, T412, T414) → U2_GGGA(T409, T410, T411, T412, T414, r1B_in_ggga(T410, T409, .(T411, T412), T414))
R1B_IN_GGGA(.(T409, T410), T411, T412, T414) → R1B_IN_GGGA(T410, T409, .(T411, T412), T414)

The TRS R consists of the following rules:

revA_in_ga([], []) → revA_out_ga([], [])
revA_in_ga(.(T24, []), .(T24, [])) → revA_out_ga(.(T24, []), .(T24, []))
revA_in_ga(.(T51, .(T50, [])), .(T50, .(T51, []))) → revA_out_ga(.(T51, .(T50, [])), .(T50, .(T51, [])))
revA_in_ga(.(T88, .(T87, .(T86, []))), .(T86, .(T87, .(T88, [])))) → revA_out_ga(.(T88, .(T87, .(T86, []))), .(T86, .(T87, .(T88, []))))
revA_in_ga(.(T135, .(T134, .(T133, .(T132, [])))), .(T132, .(T133, .(T134, .(T135, []))))) → revA_out_ga(.(T135, .(T134, .(T133, .(T132, [])))), .(T132, .(T133, .(T134, .(T135, [])))))
revA_in_ga(.(T192, .(T191, .(T190, .(T189, .(T188, []))))), .(T188, .(T189, .(T190, .(T191, .(T192, [])))))) → revA_out_ga(.(T192, .(T191, .(T190, .(T189, .(T188, []))))), .(T188, .(T189, .(T190, .(T191, .(T192, []))))))
revA_in_ga(.(T259, .(T258, .(T257, .(T256, .(T255, .(T254, [])))))), .(T254, .(T255, .(T256, .(T257, .(T258, .(T259, []))))))) → revA_out_ga(.(T259, .(T258, .(T257, .(T256, .(T255, .(T254, [])))))), .(T254, .(T255, .(T256, .(T257, .(T258, .(T259, [])))))))
revA_in_ga(.(T336, .(T335, .(T334, .(T333, .(T332, .(T331, .(T330, []))))))), .(T330, .(T331, .(T332, .(T333, .(T334, .(T335, .(T336, [])))))))) → revA_out_ga(.(T336, .(T335, .(T334, .(T333, .(T332, .(T331, .(T330, []))))))), .(T330, .(T331, .(T332, .(T333, .(T334, .(T335, .(T336, []))))))))
revA_in_ga(.(T365, .(T364, .(T363, .(T362, .(T361, .(T360, .(T359, .(T357, T358)))))))), T367) → U1_ga(T365, T364, T363, T362, T361, T360, T359, T357, T358, T367, r1B_in_ggga(T358, T357, .(T359, .(T360, .(T361, .(T362, .(T363, .(T364, .(T365, []))))))), T367))
r1B_in_ggga([], T397, T398, .(T397, T398)) → r1B_out_ggga([], T397, T398, .(T397, T398))
r1B_in_ggga(.(T409, T410), T411, T412, T414) → U2_ggga(T409, T410, T411, T412, T414, r1B_in_ggga(T410, T409, .(T411, T412), T414))
U2_ggga(T409, T410, T411, T412, T414, r1B_out_ggga(T410, T409, .(T411, T412), T414)) → r1B_out_ggga(.(T409, T410), T411, T412, T414)
U1_ga(T365, T364, T363, T362, T361, T360, T359, T357, T358, T367, r1B_out_ggga(T358, T357, .(T359, .(T360, .(T361, .(T362, .(T363, .(T364, .(T365, []))))))), T367)) → revA_out_ga(.(T365, .(T364, .(T363, .(T362, .(T361, .(T360, .(T359, .(T357, T358)))))))), T367)

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 3 less nodes.

(6) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

R1B_IN_GGGA(.(T409, T410), T411, T412, T414) → R1B_IN_GGGA(T410, T409, .(T411, T412), T414)

The TRS R consists of the following rules:

revA_in_ga([], []) → revA_out_ga([], [])
revA_in_ga(.(T24, []), .(T24, [])) → revA_out_ga(.(T24, []), .(T24, []))
revA_in_ga(.(T51, .(T50, [])), .(T50, .(T51, []))) → revA_out_ga(.(T51, .(T50, [])), .(T50, .(T51, [])))
revA_in_ga(.(T88, .(T87, .(T86, []))), .(T86, .(T87, .(T88, [])))) → revA_out_ga(.(T88, .(T87, .(T86, []))), .(T86, .(T87, .(T88, []))))
revA_in_ga(.(T135, .(T134, .(T133, .(T132, [])))), .(T132, .(T133, .(T134, .(T135, []))))) → revA_out_ga(.(T135, .(T134, .(T133, .(T132, [])))), .(T132, .(T133, .(T134, .(T135, [])))))
revA_in_ga(.(T192, .(T191, .(T190, .(T189, .(T188, []))))), .(T188, .(T189, .(T190, .(T191, .(T192, [])))))) → revA_out_ga(.(T192, .(T191, .(T190, .(T189, .(T188, []))))), .(T188, .(T189, .(T190, .(T191, .(T192, []))))))
revA_in_ga(.(T259, .(T258, .(T257, .(T256, .(T255, .(T254, [])))))), .(T254, .(T255, .(T256, .(T257, .(T258, .(T259, []))))))) → revA_out_ga(.(T259, .(T258, .(T257, .(T256, .(T255, .(T254, [])))))), .(T254, .(T255, .(T256, .(T257, .(T258, .(T259, [])))))))
revA_in_ga(.(T336, .(T335, .(T334, .(T333, .(T332, .(T331, .(T330, []))))))), .(T330, .(T331, .(T332, .(T333, .(T334, .(T335, .(T336, [])))))))) → revA_out_ga(.(T336, .(T335, .(T334, .(T333, .(T332, .(T331, .(T330, []))))))), .(T330, .(T331, .(T332, .(T333, .(T334, .(T335, .(T336, []))))))))
revA_in_ga(.(T365, .(T364, .(T363, .(T362, .(T361, .(T360, .(T359, .(T357, T358)))))))), T367) → U1_ga(T365, T364, T363, T362, T361, T360, T359, T357, T358, T367, r1B_in_ggga(T358, T357, .(T359, .(T360, .(T361, .(T362, .(T363, .(T364, .(T365, []))))))), T367))
r1B_in_ggga([], T397, T398, .(T397, T398)) → r1B_out_ggga([], T397, T398, .(T397, T398))
r1B_in_ggga(.(T409, T410), T411, T412, T414) → U2_ggga(T409, T410, T411, T412, T414, r1B_in_ggga(T410, T409, .(T411, T412), T414))
U2_ggga(T409, T410, T411, T412, T414, r1B_out_ggga(T410, T409, .(T411, T412), T414)) → r1B_out_ggga(.(T409, T410), T411, T412, T414)
U1_ga(T365, T364, T363, T362, T361, T360, T359, T357, T358, T367, r1B_out_ggga(T358, T357, .(T359, .(T360, .(T361, .(T362, .(T363, .(T364, .(T365, []))))))), T367)) → revA_out_ga(.(T365, .(T364, .(T363, .(T362, .(T361, .(T360, .(T359, .(T357, T358)))))))), T367)

The argument filtering Pi contains the following mapping:
revA_in_ga(x1, x2) = revA_in_ga(x1)
[] = []
revA_out_ga(x1, x2) = revA_out_ga(x1, x2)
.(x1, x2) = .(x1, x2)
U1_ga(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U1_ga(x1, x2, x3, x4, x5, x6, x7, x8, x9, x11)
r1B_in_ggga(x1, x2, x3, x4) = r1B_in_ggga(x1, x2, x3)
r1B_out_ggga(x1, x2, x3, x4) = r1B_out_ggga(x1, x2, x3, x4)
U2_ggga(x1, x2, x3, x4, x5, x6) = U2_ggga(x1, x2, x3, x4, x6)
R1B_IN_GGGA(x1, x2, x3, x4) = R1B_IN_GGGA(x1, x2, x3)

We have to consider all (P,R,Pi)-chains

(7) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(8) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

R1B_IN_GGGA(.(T409, T410), T411, T412, T414) → R1B_IN_GGGA(T410, T409, .(T411, T412), T414)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
R1B_IN_GGGA(x1, x2, x3, x4) = R1B_IN_GGGA(x1, x2, x3)

We have to consider all (P,R,Pi)-chains

(9) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(10) Obligation:

Q DP problem:
The TRS P consists of the following rules:

R1B_IN_GGGA(.(T409, T410), T411, T412) → R1B_IN_GGGA(T410, T409, .(T411, T412))

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(11) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

R1B_IN_GGGA(.(T409, T410), T411, T412) → R1B_IN_GGGA(T410, T409, .(T411, T412))
The graph contains the following edges 1 > 1, 1 > 2

(0) Obligation:

(1) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

(2) Obligation:

(3) DependencyPairsProof (EQUIVALENT transformation)

(4) Obligation:

(5) DependencyGraphProof (EQUIVALENT transformation)

(6) Obligation:

(7) UsableRulesProof (EQUIVALENT transformation)

(8) Obligation:

(9) PiDPToQDPProof (SOUND transformation)

(10) Obligation:

(11) QDPSizeChangeProof (EQUIVALENT transformation)

(12) YES